These methods, including the generalized-\(\alpha \)  method, are limited to the second-order accuracy in time. In contrast, high-order methods like Lax-Wendroff, Runge-Kutta, Adams-Moulton, and backward differentiation schemes (see [1]}) or high-order IGA collocation methods for dynamics (see, e.g., [2]})
lack explicit control over the numerical dissipation of high frequencies. Thus, we propose an explicit \(k\) -step generalized-\(\alpha \)  method that delivers \(2k\)  accuracy in time for second derivatives problems in time. Our method obtains high-order accuracy in the time with optimal control of the resulting system's spectral behaviour. Thus, we build an algorithm that consists of \(3k\)  equations; for each three-equation set, we solve an explicit system for a variable and update the other two. We then study the resulting amplification matrix's spectral properties to determine the CFL condition and introduce user-defined parameters that control the numerical dissipation. Later, we prove that the CFL condition of our method is independent of temporal accuracy.
